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On local flat homomorphisms and the Yoneda Ext-algebra of the fibre. (English) Zbl 0572.13002

Homotopie algébrique et algèbre locale, Journ. Luminy/France 1982, Astérisque 113-114, 227-233 (1984).
[For the entire collection see Zbl 0535.00017.]
Let f: (A,\({\mathfrak m},k)\to (B,{\mathfrak n},\ell)\) be a local flat homomorphism of noetherian local rings, with fibre \(\bar B=B/{\mathfrak m}B\). If \(\pi^*(A)\) denotes the homotopy Lie algebra of A (so that \(Ext^*\!_ A(k,k)\) is the universal enveloping algebra of \(\pi^*(A))\), it is known from work of Gulliksen [T. H. Gulliksen and G. Levin, ”Homology of local rings”, Queen’s papers Pure Appl. Math. 20 (Kingston, Ont. 1969; Zbl 0208.303)], the reviewer [Math. Ann. 228, 27-37 (1977; Zbl 0333.13011)] and M. André [Comment. Math. Helv. 57, 648-675 (1982; Zbl 0509.13007)], that there exist six-terms exact sequences of \(\ell\)-vector-spaces: \[ 0\to \pi^{2i-1}(\bar B)\to^{g^*}\pi^{2i-1}(B)\to^{f^*}\pi^{2i-1}(A)\otimes_ k\ell \to^{\delta^{2i-1}}\pi^{2i}(\bar B)\to^{g^*}\pi^{2i}(B)\to^{f^*}\pi^{2i}(A)\otimes_ k\ell \to 0 \] for every integer \(i\geq 1\), with furthermore \(\sum_{i>0}\dim Im(\delta^{2i-1})\leq e\dim \bar B-depth \bar B.\) The main result of the present paper shows Ker \(g^*\) is contained in the center of \(\pi^*(\bar B)\). As an application, the author shows that if A is attached to a regular local ring by a sequence of Golod epimorphisms, then the center of \(\pi^*(A)\) is concentrated in degrees 1 and 2, and conjectures this property holds for all local rings.
Reviewer: L.L.Avramov

MSC:

13D99 Homological methods in commutative ring theory
13H05 Regular local rings
17B55 Homological methods in Lie (super)algebras
13E05 Commutative Noetherian rings and modules
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)